the cofactor of the determinant (43), which is formed
by deleting rows containing functions kl, k2, ...,k and
columns containing functions 11 ...,1 and evaluating
1 2 p

r

d- f S IT, KLI

c brlp~~~

- 53 -

the resulting determinant and applying an appropriate

parity factor. The parity factor is -1 if the sum of
the cardinal positions of the function indices in the
original determinants is odd, and +1 if the sum is
even. With these definitions, we obtain

JW (i\.I p ) f- 21 1 2 (LI CL ** LI))

K L

The notation 4 above a summation means that all
configurations containing the orbitals in set A are to
be included. We can simplify (46) by introducing a
discrete p-matrix,

E < (< ms i l l1 U-

[^ LI C^( M LI) ^ LJ,/J

(47)

The presence of a tilde under a matrix signifies that
it is indexed by the non-ordered sets k, f. We drop
the tilde wthen we mean the part indexed by ordered sets
of indices, b and We then have

- 54 -

(?) 7 X (.. (') ('T

,K ) ~(48)

The peculiar reversed form of the indices I and J Is
simply to make the matrix form of this equation
simpler. The discrete transition p-matrix satisfies

r ---- r (49)

Pr k(j/i ) (50

where P permutes any two indices in a set. That is, it
is antisymmetric with respect to interchange of any two
indices on the same side of the bar, and is Hermitian
if I = J. The summations in (48) are over all values
of the indices. With a little study, one can use (50)
to reduce (48) to

_L 3 (i... 51)
ft p) (51)

where the summations are now over ordered sets of
indices. This is extremely important for computational

- 55 -

use, since we will only need a small part of the full

p-matrix in (48). In fact, for M orbitals, the full p-

matrix in (47) has dimension M ; the part with ordered

indices has dimension ( ) which for large M goes as
p
rP /p!.

Thus far, we have essentially followed L&wdin

/18/, except that we have used the transition matrix

throughout and introduced Slater determinants in (51).

At this point, LBwdin specialized to orthonormal

orbitals (Ref. 18, eq. 59); we shall avoid this

restriction.

Now we would like to study the properties of

the discrete p-matrix. Using (44) and (48), for I = J,

.....,..

For orthonormal orbitals, S.. = .., and (52) reduces
Ij IJ
to

^ r(P- i(V Ps Fej(P)ij (53)
'S < '

- 56 -

.z EF )(k 1)

-LK

In the general non-orthogonal case,
re r i tten

(52) can

& rF"

where

f S = s ...

is the direct product of p overlap matrices. The
product matrix in (54) is not Hermitian, which is
rather unpleasant to deal with computationally. We
note that since the trace is invariant under
permutations of the product matrices that

Lt = C(fPIRYM ']- tb(li

(55)

so we define

(54)

PiI tr ~B~

- 57 -

f7T')

Y C^'ff"

P")

fR2 'TY2

and then in the general case, we will have

ir 61(/

-/ N

From now on, we shall refer to the primed Hermitian
matrix in (56) as the p-matrix, because it has the nice
properties that hold for the case of an orthonormal
basis.
To proceed, we need some theorems on direct
product matrices, which we state here, and prove in
Appendix VII.

> 0, all rational powers r exist, and we can
compute (56) from the eigenproblem solution
basis overlap matrix. Now (56) is Hermitian,
diagonalize it:

(64)

(c17u) A lfYJJ)

... < J) = ( x ...x -U)(AlXA -- xA) (62)

T'P''

= },uI

r"' /m-~2-Y2 6'"~`"

- 59

For I = J, (48) can be written

-S X .

- 0 / x j ... x '-

S# r.... x

,a-^Os-Yj['yf--'-16 ]

(65)

where

(66)

and

r 1

(67)

There is-an infinite number of ways of orthonormalizing
a basis. It is interesting that the particular one
(66) comes into this quite naturally. It is the well-
known symnetric orthonormal ization /70,71/ which has
the particular property that if the basis set is
symmetry adapted, the orthonormalized set is symmetry
adapted also to the same symmetry operations.

Fr7 (x/x) =

[ (f)X (U .. (,] [.(ti Y (2)... 'p

0-= :y'

t(r/'c = S ^^/^ S1">

- 60 -

Substituting (63) into (65), we can bring the expansion
of the p-matrix (48) to diagonal form:

r~iui=

(68)

~i (69a)

f I (69b)

The general form (69b) is again more convenient for
computation because we need deal only with ordered sets
of indices.
Practically speaking, the operators that one
is usually interested in contain at most two-particle
interactions, so according to (41), we need at most the
2-matrix. Let us therefore specialize (68) to the
cases p = 1 and p = 2. We obtain for p = 1,

) oL J~)il

--- \ ) < 4(n'\ (70)

?I [( 6',%- c a ] (5-6-...r)dI]

where
^= ^ L

#Vz LA

and i is obtained from

and for D = 2,

r1l) 21'2')

C de (6k, 6, (1L (11
J

*^^~ z IO
rz-Pr

.,n (J tz(,) A* (1

(72)

whe re

(,,(1,2) =

(73)

The orbitais defined by (71) are cal led the natural
spin orbitals (NSO); the functions defined by (73) are
called the natural spin geminals (NSG). The p-matrix

- 61 -

(71)

I -3
7- 2j Jez^) m.
,2.

- 62 -

eigenvalues are called occupation numbers, and the

eigenvectors, natural p-states.

Obviously, had we used the orthonormal basis

(66) to begin with, the primed p-matrix in (56) would

have been obtained directly from (47) for I = J. For

this case, for the diagonal elements we find

so that

O -a _r i o f ( ) 1 (74 )

The diagonal expansion of the p-matrix in

terms of the occupation numbers and natural p-states is

particularly convenient for the computation of

expectation values. We obtain from (70) and (72)

Z < ,I >, X > (75)
K

and

- 63 -

ce/>- L Zr

; 2 A,-1o (76)

The NSO and NSG have the physical significance that

they are the set of functions for which expectation

values are strictly additive. The occupation number

factor in (75) and (76) is computationally important

because it means that the sum may usually be truncated

after the larger occupation numbers.
The discussion of the eigenproblem has thus

far been centered on the non-trans'ition p-matrix case.

To the author's knowledge, no work has been published

on the transition matrix eigenproblem. In fact, Bingel

and Kutzelnigg (See ref. 67) seem to be the only

authors who have carried out derivations in terms of

the more general transition p-matrix.
As we noted earlier, according to (49), the

transition matrix is not Hermitian. It is well-known

/73/ that an arbitrary matrix may be brought to triangular
form T by a similarity transformation with a non-singular

matrix :

W TW= T (77)

The eigenvalues of Fare the diagonal elements of A.
The eigenvectors are found as follows.

J r WT Vh'

t 1 (78)

This is a triangular set of equations for the vector
JV from which V may be determined according to

VV') (79)

Since we do not have a diagonal form of the matrix ,
the simple results (75) and (76) do not hold. There is
consequently no value in determining eigenvalues and
elgenvectors of the transition p-matrix because doing
so does not simplify the situation. Instead, one

- 64 -

- 65 -

simply computes transition values directly from (41)

and (51):

^>^' Qo frI4>( rl}~4

< Z

lur 1i

J,
ii

(80)

Thp RPnduired Density Matrix for a lon-CI

Wavefunction

If the wavefunction does not have the Cl form
(42), one can still obtain a reduced density matrix
front (34). Obtaining a representation of it in a
discrete basis is not difficult. The p-matrix may be
considered the kernel of an operator such that

f F (x/x, ,)/'xdx' /

(81)

for arbitrary functions
eigenfunction of then

f(x).

If f(x) is an

1 _X ~__~_Y___ ______ _I _~_ _~_ _

(Jz, P 0

F' f() =

- 66 -

r"F ? ANW

(82)

Larsson and Smith /74/ have recently used this relation
to derive NlSO's of the 1-matrix of Larsson's Hylleraas-
type wavefunction for lithium. They introduce an M-
function basis and expand the NSO's in this basis:

(83)

Using this in (82) gives

(8 i)

which leads to the secular problem

C Cr< ^ ~

(86)

where

(85)

? = 4 C

a~-~~

IC sck

- 67 -

- f e)f (y ) k 4& (87)

J4. ,
--

f07 (x) C&

(88)

The secular problem (86) may be solved by the methods
discussed in Appendix V.
The occupation numbers calculated this way
will be lower bounds to the exact occupation numbers.
The proof is not difficult. Let the set of exact
normalized ISO be with eigenval.ues A :

9',% K- A

(89)

Now let us order the exact and approximate solutions
according to

14z 7

>- 0

(90)

and then construct the operators

- 68 -

9x, 21 (91)

P E 2 IZ<>1 / (92)
k.-M+/

which are projection operators satisfying the usual

relations. From the theory of outer projections /75/,
we know that for an arbitrary projection operator 0,

and any operator I bounded from below, the eigenvalues
of O0O are upper bounds in order to those of /.

Now oSO= 6 (93)

and 01 O i (Z' /)

so () n^ -z v : (94)
so that 9j has eigenfunctions X and eigenvalues and

we have immediately

,i /i (95)

The approximate occupation numbers are therefore lower

bounds as stated. The sum of the approximate
N
occupation numbers approaches ( ) from below and provides
a convenient measure of the adequacy of the chosen
basis.

- 69 -

2.3 Properties of Density latrices

Cl Expansion Convergence

In Chapter I, we mentioned the convergence

problem in the CI method. In his original paper on

density matrices, L8wdin /18/ showed that the natural

spin orbitals are actually the orbitals which give the

most raid convergence of the Cl expansion, the HSO of

highest occupation number being the most important. Of

course, one needs to know the wavefunction to begin

with in order to obtain the p-matrices and the natural

p-states. However, if a truncation is made of the CI,

one can obtain NSO for this truncated function, put

these back into a new CI, perform a new truncation

based on the size of the NSO occupation numbers or

other criteria, obtain new NSO, and so on. This

natural spin orbital iteration technique has recently

become quite a popular tool in Cl calculations, but the

convergence of the scheme does depend on the quality of

the initial truncation.

- 70 -

Bounds on Occupation Mumbers

Since the p-matrix is positive and of finite

trace, its eigenvalues obviously satisfy

o ~ ()

(96)

Coleman (See Ando, ref. 66) showed that

(97)

p = 1 and p = 2 takes the form

OLX

(2)
O"i1L

L Ni ( )

1 CMn> !)

A/L ( >3)
(,'>

it can be shown that the upper bounds are never

attained except for p = 1 and p = N 1. Sasaki /76/

obtained better bounds than these, the first few of

which are

which for

(98)

(99)

- 71 -

;11

21. '2)

;1
'3

(100)

_ 1+ 3.[i V-3)]

where [x] is the integral part of x.
He also proved that the bound for p = 2 is the best
possible.

The Carlson-Keller-Schmidt Theorem

Carlson and Keller /77/ showed that the non-
zero eigenvalues of the p-matrix are identical to those
of the (N-p)-matrix, and if the number of non-zero
eigenvalues is finite, then these two matrices are
unitarily equivalent. In addition, if

and

p (Ac~l~f(jt 21 c (i ~ 1G~ry) i

(101)

1,11~4

. 1 --

,/ (102)

- 72 -

then

",^-A (ovfSY )^ (104)

and

7 y) (A/ (105)

If and were derived from an antisymmetric wavefunction,

then the resolution (103) of the wavefunction is

automatically antisymmetric already. The

eigenfunctions of the p-matrix are called natural p-

states, and those of the (N-p)-natrix, co-natural p-

states. Coleman /65/ later pointed out that this

theorem had already been discovered more than fifty

years earlier by Schmidt /7'/. Schmidt's results, in

the terminology of density matrices, show that the

expansion (63) gives optimal convergence in the least

square sense to the wavefunction; this, coupled with

the fact that the natural n-states can always be

expanded in terms of the ISO /65/, leads to the CI

convergence theorem independently obtained by L8wdin

which we referred to earlier.

- 73

The Carlson-Keller-Schmidt theorem is of

particular significance for N = 3, since the 1- and 2-

matrices then have identical non-zero eigenvalues, and

the NSO and NSG can be obtained from each other by

virtue of (104) and (105).

Symmetry Pronerties

We mentioned earlier that the wavefunctions

should be required to be eigenfunctions of the group of

the Hamiltonian, and the question of how the symmetry

properties of the wavefunction carry over to the p-

matrices and the natural p-states has been extensively

studied. We shall merely list some of these results

here which have significance for our own work.

Theorem 1: If 1 is an N-electron Hermitian operator

of the form

4"- z (106)

or a unitary operator of the form

.n j=4 (107)
-I

or an antiunitary operator of the form

- 74 -

i=/
(Al) IT fi(108)

where, in (107) and (108), R ;is unitary

and K denotes complex conjugation, and

if P is an eigenfunction of then

the natural p-states can be chosen as

eigenfunctions of .

Theorem 2: If 4 and J transform as the irreducible

representations 9 and d respectively of

some group, then /7 transforms as the

direct product representation ex X

The particular significance of these results

is best illustrated by a few examples. If the

wavefunction is an eigenfunction of L S or rarity,
z z
theorem 1 applies, and the p-matrix blocks by ML, M'S

or parity value, and the natural n-states are

eigenfunctions of L S or parity. If the wavefunction
9 2
is an eigenfunction of S2 or L2, the natural p-states
2 2
can generally not be chosen eigenfunctions of S or L

except when S = MS = 0 or L = ML = 0. Of course, for

special choices of approximate wavefunctions,

additional symmetries may be introduced. Garrod has

shown for example that if the wavefunction is taken as

- 75 -

an average of M components with identical space and spin

parts, then the NSG's can also be made eigenfunctions

of L2. Theorem 2 is probably more useful for molecules

and solids; for atoms it essentially duplicates theorem

1.

In order to better see the structure of the

1- and 2-matrices, it is sometimes useful to expand

them in terms of separated space and spin parts. If

the wavefunction is an eigenfunction of S it may be

shown that

l^')c o 9 i

,-~ ,-~

(109)

F 61 W'~~i:7bd

Scd / ./rr, cc

, dc/^' rj-dr,

dc I /r, dc

(110)

where

a-c d

C r -I{CLA^^CL)

(111)

dJ L (c-e
_V-2 Ig -sc

The presence of the cross terms cd* and dc* in (110)

shows that the 2-matrix is generally not an

F Cx, 'iY/2

- 76 -

2
eigenfunction of S2. Also, one sometimes introduces the

charge-density 1-matrix,

(112)

the spin-density 1-matrix,

?d (riy)

(113)

zfS'z6- I/'C')

and the chare-density 2-matrix

and the charge-density 2-matrix,

(114)

The eigenfunctions of the charge-density matrices are
called charge-density natural orbitals (CDNO) and
charge-density natural geminals (CDM'G), or simply,
natural orbitals (NO) and natural geminals (NG). The
eigenfunctions of the spin-density 1-matrix are called
spin-density natural orbitals (SDIO). To find bounds

on the eigenvalues of (112) and (113), we can use the
matrix representations

S(^ ')- = f I )d6;
a ^ a ^

fo; r6i C ) cla- r'

I~~vEi ~rL ;j

- 77 -

dr7 c

t P/I.

(115)

(116)

We then use the result that if matrices /4, and C have
eigenvalues a,, /, and respectively, arranged in non-
increasing order, and if

--= /f *(117)

then /73/

nnu- (L //4/ c// A,/A

(118)

From this result, we obtain the following bounds.

0 x40- x M7.2 "i>)

l\m~ ^:;jc. I )~~ ~ S d

-fc, /)Z 1

(120)

The interest in the CF!O and CDING is two-fold. First,
if the wavefunction is an eigenfunction of S2 and S
and if MS = 0, then nd are identical, and the NSO's
are NO's with l or spini; F and /'vanish, and the I!SG's
are NG's with one of the four spin functions (111).

,,,_ ~

75' --

- 78 -

Second, for an t-function basis, there are 2M NSO, only

ft of which can be spatially linearly independent.

Consequently, the Il linearly independent NO's have

sometimes been suggested for the CI iteration scheme

discussed earlier. In general, both the NSO's and N!O's

will have mixtures of either odd or even values of

angular momentum; that is, s orbitals will have s, d,

g, i, ... admixture, and p orbitals, f, h, j, ...

admixture, and so on. This mixing poses a

computational difficulty in that most programs are set

up to deal with orhitals of a single (l,ml) value rather

than of a single mi value; angular-momentum projections

become considerably more involved if 1-mixtures are

allowed.

Ihile we have not made explicit use of them,

we have generated the CDHO's and SDNO's for all the

systems studied in this work.

Density Matrices of Some Special Functions

For a single Slater determinant of N

orbitals, the 1-matrix has N occupation numbers equal

to 1, and the remainder equal to 0. If the orbitals

are orthonormal, the 1-matrix is diagonal directly from

- 79 -

(46), and the orbitals are the NSO. This is a
particularly important case and has been discussed
extensively by L8wdin /18,19,20/. In this connection,
it is worthwhile to introduce the extended Hartree-Fock
(EHF) equations which LBwdin derived for an orthonormal
basis set. We mentioned these briefly in the last
chapter, but deferred a derivation because the density
matrices provide a particularly convenient tool for
this. We begin with the expression (41) for K = L,
where Q is now the Hamiltonian operator.

> = Ho

* J.h-/, ( )1 ,

9. rwd, ..

(121)

Varying the expression (51), we obtain

S z

yi'^ ..- ) s iPrWk) 7 /

, ,. (ii.%.. p &

. c ... c i 0

+ Complex coyj'j._ e. (122)

- 80 -

Using this result, we find

So t d et )

where we have introduced a Hermitian matrix of Lagrange
multipliers to maintain orbital normalization. By the

usual argument, the expression in brackets must vanish;

we then multiply by 8() and sum over Y obtaining the
EHtF equations:
EHF equations:

- 81 -

P, 1iir) < .j^ r [7 d'), > ...

+ f P ,,....p ^ ('< ...rl ,,z,... ^') d,.. dx

w= .(( (124)

where

(l/'J = /l ) < } (125)

Note that nowhere have we assumed an orthonormal basis

or a particular form of the orbitals; unless we start
with an orthonormal basis, there is no need even to
introduce the Lagrange multipliers, and the right-hand

side of (124) then vanishes. For a non-orthonormal
basis then, there is no need to determine Lagrange
multipliers, but we have a more difficult p-matrix to
compute. In general, it is not possible to

simultaneously diagonalize the 1-matrix and the
Lagrange multiple ier matrix, so we essentially lose the
concept of orbital energies.
It is often useful to introduce a quantity,
called the "fundamental invariant", defined by

- 82 -

p 0

which satisfies

='

where M is the number of orbitals in the basis.
invariance follows from the fact that a nonsi

linear transformation on the basis leaves
unchanged:

IrA> T V' < >Tk TI

-- /

(129)

For the case M = N, L8.wdin /19/ showed that o determines
all the p-matrices, and these are given explicitly by

(126)

(127)
(128)

The
ngu ar
(126)

< ^ >'

! > '+14>' 4

- 83 -

S= J (/...(p,)) (130)

The fundamental invariant therefore contains all the

information contained in a single-determinant

wavefunction, regardless of the form of the basis

orbitals. This noint has lead to some confusion in the

literature. In an often-quoted paper, Bunge /44/

arrived at the result that for a PGSO wavefunction, the

EHF equations do not yield unique orbitals; i.e. the

fundamental invariant is not invariant. This result is

incorrect; the error in the paper is the omission of

the factor <>> in ,p ; this simplified form holds for an

orthonormal basis. Bunge then proceeded to vary the

orbitals, destroying the orthonormality. The EHF

equations are perfectly well-defined, even for GSO.

That p determines all the p-matrices for a single

determinant is true, even for a Drojected determinant,

OD. The occupation numbers are 1 and 0 if OHO is

considered the modified Hamiltonian and D the

wavefunction; however, if OD is considered the

wavefunction, the occupation numbers are in general not

0 and 1 because the projection introduces new orbitals.

In this case, the fundamental invariant must be

constructed from the complete set of orbitals,

including all the ones introduced by the projection.

- 84 -

The p-matrices must still be determined by the

fundamental invariant, but the form of the natural p-

states and occupation numbers is not obvious. For the

case of a spin-projected determinant of pure spin

orbitals, Harrinan /79/ has derived explicit formulae

for the 1-matrix, NSO, and occupation numbers,

Hardisson and Harriman /80/ derived a formula for the

2-matrix, and this has recently been extended to point-

group and axial-rotation symmetry projection by Simons

and Harriman /81/ to obtain formulae for the 1- and 2-

matrices. The first two /79,80/ are derived for a

projected DODS determinant; in the last /81/, the

orbitals are only assumed to be orthonormal. The form

of the occupation numbers and the natural p-states for

a PGSO wavefunction is not known in analytic form,

although we have calculated the 1- and 2-matrices

directly from the projected determinant treated as a CI

expansion over non-orthonormal orbitals. The formulae

for the DODS case are already very complicated; in view

of the great increase in complexity in going to GSO, we

feel that an attempt at obtaining an analytic formula

for the p-matrices of a PGSO wavefunction would not be

worthwhile.

- 85 -

2.4 The N-Representability Problem

The Schr8dinger equation, (1), has never been

solved exactly for a system with more than one

electron. As the number of electrons increases, the

approximate wavefunctions become increasingly complex.

The Hylleraas coordinate functions discussed in the

last chanter have not been extended beyond four-

electron systems; the CI programs mentioned are limited

to less than forty electrons. Systems of chemical

interest frequently have hundreds or thousands of

electrons which we have so far been unable to treat

accurately. One can imagine Colenan's excitement in

1951 when he first observed the significance of the

equations (41) and (121); since the usual Hamiltonian

employed contains at most two-particle terms, the

energy, and all one- and two-electron properties defend

at most upon the 2-natrix, from which the 1-matrix can

be derived according to (36). The 2-matrix is a

function of only four particles. Thus, by varying a

certain four-particle function, one should be able to

obtain practically every result of chemical interest

for any system, no matter how large. Rather than

launch a calculation on DNA, Coleman contented himself

at that time with a calculation on lithium, a three-

electron system. The calculation gave an energy 30%

- 86 -

below the experimental value, in seeming violation of

the Rayleigh-Ritz variational principle. The

difficulty was that the four-particle function had been

varied over too wide a class of functions. This

problem has since become known as the "N-

representability" problem -- the problem of finding the

conditions under which a 2p-particle function, such as

a p-matrix, can be shown to be derivable from an N-

particle antisymmetric (or symmetric) wavefunction

without actually exhibiting that N-particle function.

This problem has received a great deal of study in the

last two decades. The indications so far

pessimistically are that either the general solution

does not exist, and therefore cannot be found, or that

if the solution exists, and is found, implementing it

will be at least as difficult as carrying out a

calculation with the N-particle wavefunction. This

thought is rather depressing, considering that a

feasible solution has the strong possibility of

revolutionizing a good part of chemistry, physics, and

biology. More optimistically, one might hooe for an

approximate solution so that variation of a reduced

density matrix could be implemented in such a way as to

provide a useful alternative to ab initio, semi-

empirical, or even empirical theories. Some progress

87 -

has been made along these lines by a number of authors

/82 90/.

In the meantime, reduced density matrices

provide a convenient tool for interpretation of

wavefunctions and nronerties.

CHAPTFR 3

ATOMIC PROPERTIES

3.1 Introduction

Reading the current quantum chemical

literature gives one the feeling that a total energy is

the only property atomic and molecular systems possess.

Since the total energy, like the thermodynamic enthalny

and free energy, is meaningless except when compared

with another total energy, one miFght even begin to

question the motivation of the calculations. In fact,

of course, there are a good many oronerties of interest

which we can in orinciole compute. A recent hook by

Mfalli and Fraga /92/, although somewhat concise and

uncritical, does at least give an idea of some of the

orooerties of interest. A review article by Doyle

/105/ discusses relativistic and non-relativistic

corrections to atomic energy levels and a number of

numerical tables with these corrections is given. We

will content ourselves in this chapter only with givinr

a short indication o4 some of these properties with

- 88 -

references to work where greater detail may be found.

3.2 Enprries

The calculation of the energy determines the

wavefunction. Fxcept for one-electron systems, which

can be solves exactly, and Pekeris' work cited earlier

on two-electron systems, calculations of energy levels

cannot corrnte with exoerimnnt in accuracy.

Consenuently, xr-ent for determination of the

wavefunction and com)nrison with other theoretical

results, for atoms, calculation -of energies is of

little interest because the exnerimental data is so

much better. Cor nolocules, even small di-tomics, this

is not the case, and one can often cet better

characterization of nntential curves by theoretical

comnutat ions than current exner mental n thds can

five. It is perhaps one of the sad facts of quantum

mechanics that determination of the energy is the only

route to the wavefunct ion, ann that even if an

aonroximate wavefunction gives a good enerTy, other

properties calculated from it may be rather Door.

In this work, in addition to the energy

determination, we have also evaluated the scale factor

- 89 -

- 90 -

and scaled energy given for atoms by /98/

(131)
.2

E V< > __ (132)
2 4 =c-'/

An atomic wavofunction may always be scaled to satisfy

the virial theorem; if the unsealed wavefunction

satisfies it already, then the scale factor is

necessarily unity. We have found this useful in that

a scale factor differing from unity by more than about

0.001 indicates that the basis is poorly chosen.

3,3 Snecific Mass Fffect

In thp introduction, we derived the snpcific

mass effect, or mass nolarization, correction to the

kinetic energy, en. (15). FrmHan /99/ has estimated
-6
the effect from Pxnprimental snectra to be about 10

!1 (0.2 K) for Li 2 S and 10-7 H for Li 4 2S. He also

states that tho effect should be approximately

independent of Z, so that the same estimates apply to

the rest of the isoelectronic sequence. However,

- 91 -

Prasad and Stewart /100/ have recently evaluated the

effect from Weiss' 45-term CI wavefunctions for the 2

S and 2 2P states of the sequence from Z = 3 to 8; for

the 2 2S state, their data gives the shift proportional

to 1.29; or the 2 2P states, the shifts decrease with

increasing Z, becoming negative for Z > 4. The shift
9 7 6
for the 2 'S state of Li is 2.587 K, and for Li6, 3.017